Witt vectors of non-commutative rings and topological cyclic homology

Classically, one has for every commutative ring A the associated ring Wn(A) of p-typical Witt vectors of length n in A. We extend this construction to a functor which assigns to every assiciative (but not necessarily commutative or unital) ring A an abelian group Wn(A). The extended functors comes equipped with additive restriction, Frobenius, and Verschiebung operators. We write W(A)F for the quotient group of coinvariant for the Frobenius operator. Let K*(A; Zp) denote the p-adic K-groups of A, that is, the homotopy groups of the p-completion of the spectrum K(A). We prove that, if A is a finite dimensional associative algebra over a perfect field k of positive characteristic p, then there is a canonical isomorphism
       Kq(A;Zp) = (Lq+1W(-)F)(A),
where the left-hand side is the q+1th left derived functor in the sense of Quillen of the functor W(-)F. The proof is by comparison with the topological cyclic homology TC*(A;p) introduced by Bokstedt-Hsiang-Madsen.

The original paper contained two mistakes. These mistakes, however, do not affect the main conclusions of the paper. Please see the erratum below for a correction.

Lars Hesselholt <larsh@math.mit.edu>